The small brass plaque reads B Rigby and G Peller, corsetières. The discreet sign once hung on a nondescript doorway in London's South Molton Street and marked the threshold to a secret world of ribbons, lace and stays that was for women's eyes only.

The nameplate, now in storage, is also a small piece of history, a reminder of one Jewish woman's flight from the horror of the second world war. Last week Rigby & Peller, the now-famous lingerie business founded by the nimble-fingered Bertha Rigby and Gita Peller, was sold in a deal that valued the company at nearly £10m. Both the founders are long dead but the sale is a footnote to a rich history intertwined with the immigration that shaped London's rag trade in the last century, as well as the latest throw of the dice in the fickle world of fashion.

Little is known about the enigmatic Peller, who fled her native Hungary, legend has it, with samples of her intricate work stowed in her suitcase. June Kenton, the matriarch of the Jewish family that took over the business in the early 1980s, doubts she would have given up precious space in her suitcase for the samples: "She left [Hungary] very late. Those who escaped in 1939 left an awful lot behind."

Peller was taken in by Rigby when she arrived in Britain, and the duo established a company that, over the past 70 years, has become recognised as the gold standard in lingerie. Its famously "proper" fitting service has a formidable reputation, while rigorous levels of discretion have won it the patronage of several generations of the royal family including Princess Margaret and Princess Diana. Margaret stepped out only in bespoke R&P, while Diana mischievously took away lingerie posters for her teenage sons' walls at Eton, reveals Kenton. It has also served Hollywood stars: actresses such as Scarlett Johansson and Gwyneth Paltrow are numbered among the glamorous clientele that has relied on R&P's oomph to attain red-carpet perfection.

Tailoring and dressmaking were the most common occupations of the Jewish immigrants who huddled together in the East End of London at the start of the last century. "Tailoring was traditionally a typical Jewish occupation in the small towns and industrial cities of eastern Europe and, subsequently, immigration countries such as Britain and the US," says Sarah Harel-Hoshen, curator of the Jewish Museum in London. One estimate is that five out of seven male immigrants worked in the ready-to-wear clothing workshops that mushroomed in the area at that time. Single women worked alongside the men and after they were married would help in their husbands' workshops. The community's rich skill base included millinery, hosiery and corsetry, with the products sold from shops as well as stalls on large street markets, such as that in Petticoat Lane.

There are no exact numbers, but according to the Jewish Museum the biggest wave of immigration took place between 1880 and 1905, when between 200,000 and 450,000 Jews arrived in the UK following the pogroms and restrictions in the Russian Empire's Pale of Settlement, the area Jews were allowed to live in. That flow was staunched by the Aliens Act of 1905 but the rise of fascism saw another 50,000 Jewish refugees, including Gita Peller, arrive from Austria, Germany and central Europe in the 1930s. "Many of those in the clothing industry, who started as sweat-shop workers or very poor artisans, did well and after one or two generations – if the family remained in the tailoring or textile business – moved their businesses to the West End," explains Harel-Hoshen. In the West End the focus was on the more exclusive custom-made tailoring trade, skills that would overlap with the demands of theatreland, with firms such as Berman & Nathan and Angels gaining prominence.

Rigby & Peller has also played its part in the showbusiness tradition, by providing the corsetry that underpinned The Two Ronnies' costume extravaganzas in the 1970s and 1980s, as well as the antics of Dick Emery and Benny Hill.

A rummage through the Rigby & Peller archive, which is stored in silk-lined suitcases in an office above its Conduit Street store in Mayfair, shows that a lot has changed since Vogue coined the term "brassiere" in 1907. Kenton fingers the fine stitching on one of the pieces, a scrap of innocent-looking pink ribbons and lace – with no wires – that would truss only the daintiest of frames. R&P's stores still channel that old-fashioned glamour, with swagged curtains and thick red carpets patterned with gold, but the company has consciously moved with the times ushered in by upmarket, risqué rivals such as Agent Provocateur. Indeed, in the video for her recent single Alejandro, Lady Gaga is vamping it up in a pair of Rigby & Peller lace pants, and not much else.

Kenton reminisces that her mother had "sitting down" and "standing up" corsets, with the latter negating the possibility of the former. A romp through bra history sees the advent of adjustable straps and the creation of cups in the 1920s, the sanctity of which, in the days of eye-popping Wonderbra cleavages, Kenton is keen to preserve. "You were born with two," she says. "And as far as Rigby & Peller are concerned, those two should be in the cups. They shouldn't meet in the middle." The 75-year-old is an authority on bras, and when she says, "I dream about bras, I breathe bras, I live bras", you tend to believe her. Kenton can tell women their bust size without using a measuring tape, a skill that must save blushes at the palace – but when she narrows her eyes and promises "to get her hands on you", it is clear that you have come up wanting.

The 1930s saw the mass-production of bras with different cup sizes for the first time, but despite the competitive threat and the hardship caused by the war, Rigby & Peller's business thrived thanks to a loyal and well-heeled customer who sought out that easily missed doorway in the West End. As the cold war ushered in jutting, cone-shaped bras, the business passed to Peller's cousin, who continued its bespoke service despite the industry's own nuclear threat: the Wonderbra, which rose to prominence, so to speak, in North America in the late 1960s.

By the 1980s the lingerie market was a world away from the hushed environs of the corsetière's workshop; it was big business and the majority of women had turned against bespoke lingerie and the deft hands that could magically create an hourglass figure.

When Kenton and her husband Harold got the opportunity to buy Rigby & Peller in 1982, the business was struggling. It was still just a workshop, with four corsetières, that the two women had established above the shop in South Molton Street 40 years earlier. The writing was on the wall for bespoke lingerie, and though her husband was against buying R&P, Kenton got her way, acquiring the firm for the princely sum of £20,000. "It was too old-fashioned for most people," she says. The couple made it profitable by combining R&P with the lingerie business they already owned, and introducing off-the-peg bras and an own-label range.

Fashion also came to the rescue, as corsetry and the 1950s shape enjoyed a renaissance, in no small part down to Madonna strutting around in a pointed-cup bra at the Cannes film festival in 1991.

Today Rigby & Peller has seven stores and recorded sales of £10.1m last year. Its bras sell for between £50 and £100, although a bespoke version costs up to £200. Last week the Kentons sold an 87% stake to Belgian bra-maker Van de Velde for £8m.

Despite the sale, June and her son David, who runs the day-to-day operations, are staying on as directors. "It will be business as usual," says Kenton, adding: "When we open new stores I'll be running around doing fittings. Like us [Van de Velde] are a family-oriented company and appreciate the value of heritage."

Throughout Rigby & Peller's long history, the only relatively straight line in an industry obsessed with curves records the expansion of the British bust, a phenomenon attributed to factors such as diet, exercise and the contraceptive pill.

"In my mother's day, if you were bigger than a C, you'd have to have something made for you," says Kenton. "When we first got D and DD in 1970s, there was a waiting list. Today we go up to an N!"